As many people, I have to deal with a lot of numbers all the time (bug numbers, changelist numbers, tracking numbers, ...) and as a mathematician I sometimes notice when numbers have peculiar properties.

For example, I recently ran into 152463, which is a straight (made up of consecutive digits, not necessarily in order.) I then ran into 35426, which is also a straight. Is this odd?

To put it another way - what proportion of numbers are straights?

We consider only non-negative integer numbers {0, 1, 2, ... }. No duplicate digits are allowed, so the very last straight is 9,876,543,210.

Break it down by the length of the number in digits. For example, how many four-digit numbers are there? We can choose any number from 1-9 for the first digit, and any number from 0-9 for each of the other three digits; so there are 9 * 103 numbers of length 4.

How many of these are straights? There are seven ways to choose a set of four consecutive digits to make up a four-digit straight: {0,1, 2, 3}, {1, 2, 3, 4}, {2, 3, 4, 5}, {3, 4, 5, 6}, {4, 5, 6, 7}, {5, 6, 7, 8}, {6, 7, 8, 9}.

Each of these seven ways can then be shuffled to make the straights themselves. For six of these seven sets of digits, all possible shufflings are allowed; there are 4! shufflings for each set. This gives 6 * 4! straights.

But for {0, 1, 2, 3}, we have to be careful not to end up with 0 as a leading digit. We choose the leading digit first out of the set {1, 2, 3} (there are three ways to do this) and then we have complete freedom to shuffle the three remaining digits (including 0.) This gives 3 * 3! straights.

So there are 6 * 4! + 3 * 3! straights which are four digits long.

Considering all possible lengths from 1 to 10 (we'll consider 0 as a number of length 1) gives the following table:

Length

Numbers

Straights

% Straights

1

10

10

100.000%

2

9 * 101

17

18.889%

3

9 * 102

46

5.111%

4

9 * 103

162

1.800%

5

9 * 104

696

0.773%

6

9 * 105

3480

0.387%

7

9 * 106

19,440

0.216%

8

9 * 107

115,920

0.129%

9

9 * 108

685,440

0.076%

10

9 * 109

3,265,920

0.036%

Total

1010

40,91,131

0.041%

Since roughly 1% of five-digit numbers are straights, it is not surprising that I see them that often.

That's not enough; it is also necessary to poke a hole in the firewall to allow inbound connections. I use an indirect string for the Group name, to allow for installing localized builds. This points to the "Remote Desktop" feature group.